Equivariant liftings in Lipschitz-free spaces

TL;DR

This paper investigates conditions under which Banach spaces can be linearly lifted into their Lipschitz-free spaces in a way that respects group symmetries, with applications to complex Lipschitz-free spaces.

Contribution

It establishes the existence of equivariant linear liftings for certain groups and spaces, extending the understanding of symmetry-preserving embeddings in Lipschitz-free spaces.

Findings

Existence of G-equivariant liftings for compact groups

Extension to unions of such groups with complemented Lipschitz-free spaces

Development of a complex Lipschitz-free space for subsets of complex Banach spaces

Abstract

We consider Banach spaces $X$ that can be linearly lifted into their Lipschitz-free spaces $\mathcal{F}(X)$ and, for a group $G$ acting on $X$ by linear isometries, we study the possible existence of $G$-equivariant linear liftings. In particular, we prove that such lifting exists when $G$ is compact in the strong operator topology, or an increasing union of such groups and $\mathcal{F}(X)$ is complemented in its bidual by an equivariant projection. As an example of application, we define and study a complex version of the Lipschitz-free space $\mathcal{F}(X)$ when $X$ is a subset of a complex Banach space stable under the action of the circle group.